Show commands:
SageMath
E = EllipticCurve("hn1")
E.isogeny_class()
Elliptic curves in class 62400hn
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 62400.ho4 | 62400hn1 | \([0, 1, 0, -31233, -44366337]\) | \(-822656953/207028224\) | \(-847987605504000000\) | \([2]\) | \(983040\) | \(2.1194\) | \(\Gamma_0(N)\)-optimal |
| 62400.ho3 | 62400hn2 | \([0, 1, 0, -2079233, -1144142337]\) | \(242702053576633/2554695936\) | \(10464034553856000000\) | \([2, 2]\) | \(1966080\) | \(2.4659\) | |
| 62400.ho2 | 62400hn3 | \([0, 1, 0, -3743233, 944177663]\) | \(1416134368422073/725251155408\) | \(2970628732551168000000\) | \([2]\) | \(3932160\) | \(2.8125\) | |
| 62400.ho1 | 62400hn4 | \([0, 1, 0, -33183233, -73585358337]\) | \(986551739719628473/111045168\) | \(454841008128000000\) | \([2]\) | \(3932160\) | \(2.8125\) |
Rank
sage: E.rank()
The elliptic curves in class 62400hn have rank \(0\).
Complex multiplication
The elliptic curves in class 62400hn do not have complex multiplication.Modular form 62400.2.a.hn
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
